Limits & derivatives
Limits, the derivative as slope, and the power rule.
Learn with Kutty
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The derivative tells how fast something is changing right now.
Look: The dashed ruler touches y = x² at x = 2. It is as steep as 4.
✏️ Your turn!
Type your answer and press Check. Kutty will tell you if it is right.
1Derivative of x⁵?
2Steepness of y = x² at x = 1?
3Derivative of 3x⁴ − 5x + 7?
! Careful!
✗Derivative of x³ is 3x³.
✓Bring the power down and lower it by 1: 3x².
✗Derivative of 7 is 7.
✓A flat line does not change: 0.
See the whole lesson on one page
The derivative tells how fast something is changing right now.
On a curve, place a ruler so it just touches at one point (a tangent). Its steepness is the derivative there. For x², the derivative is 2x.
Think of it like thisA car’s speedometer shows your speed at this exact moment, not your average for the trip. That moment-speed is a derivative.
- Limit
- The value something gets closer and closer to.
Solved together
How steep is y = x² at x = 3?
The rule for x² is: steepness = 2x.
Put in x = 3.
2 × 3 = 6
AnswerSteepness 6 at x = 3.
For older learners and grown-ups: a little more
Formally f′(x) = lim (h→0) [f(x + h) − f(x)]/h. For x²: [(x + h)² − x²]/h = 2x + h → 2x. Limits like (x² − 9)/(x − 3) as x → 3 equal 6 after cancelling (x − 3).
Now practise it
Kanakku makes new questions on limits & derivatives every time. It shows the working and gives a hint when you are stuck.